They tell the doctor that doctor-patient confidentiality expires with the death of the patient. This is not the case. Confidentiality continues equally after death except for exceptions under law, which would include identifying her. Only the executor or next-of-kin could waive it.
Lt. Tau says the steps to the church are a crime scene, and Andy says they have to collect hair and fibers, but the murder did not happen on the steps, only the the casket was dropped revealing the body of the dead woman, so there is absolutely no forensic evidence to be found at the scene.